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Hardness Amplification of Optimization Problems

Authors Elazar Goldenberg, Karthik C. S.

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • hardness amplification
  • average case complexity
  • direct product
  • optimization problems
  • fine-grained complexity
  • TFNP

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Abstract

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products.
We say that an optimization problem Π is direct product feasible if it is possible to efficiently aggregate any k instances of Π and form one large instance of Π such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the k smaller instances. Given a direct product feasible optimization problem Π, our hardness amplification theorem may be informally stated as follows:
If there is a distribution D over instances of Π of size n such that every randomized algorithm running in time t(n) fails to solve Π on 1/α(n) fraction of inputs sampled from D, then, assuming some relationships on α(n) and t(n), there is a distribution D' over instances of Π of size O(n⋅α(n)) such that every randomized algorithm running in time t(n)/poly(α(n)) fails to solve Π on 99/100 fraction of inputs sampled from D'. 
As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium.

Cite As Get BibTex

Elazar Goldenberg and Karthik C. S.. Hardness Amplification of Optimization Problems. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.1

Author Details

Elazar Goldenberg
  • The Academic College of Tel Aviv Yaffo, Israel
Karthik C. S.
  • Tel Aviv University, Israel

Funding

  • Karthik C. S.: Most of this work was done while the author was a student at Weizmann Institute of Science and was supported by Irit Dinur’s ERC-CoG grant 772839. This work also received support from the Israel Science Foundation (grant number 552/16) and the Len Blavatnik and the Blavatnik Family foundation.

Acknowledgements

We would like to thank Amir Abboud, Irit Dinur, and Eylon Yogev for discussions and comments.

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